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Rationality, Expected Value, and Pascal's Wager

There's been a lot of talk about probability
in the context of Christian belief these days. From academic textbooks and
journal articles, to message boards and blogs (including this one), material
abounds on the probability of God's existence, the probability of miracles
generally, and the probability of the resurrection of Jesus given certain facts
of history in the light of Bayes' theorem. The guiding assumption behind all
this, though not always explicitly stated, is that accepting the more probable
hypothesis results in our holding the most rational belief. And most of us want
to be rational.

In a recent post here it was argued that the probability of the
resurrection of Jesus, given the specific evidence and general background
knowledge bearing on the case, is high relative to competing hypotheses. At the
least, I suggested, the resurrection should be considered not improbable (hence
not irrational to believe). But some people objected that without specific
calculations that suggestion is too vague. So now we will "plug in"
some numbers. Recall again that per Bayes' Theorem, the probability of a
hypothesis H, given evidence E and background knowledge K, equals the
conjunction of its explanatory power and prior probability:

P(E│H & K) x P(H│K)

P(H│E & K) = -----------------------------

P(E│K)

If we estimate .25 to indicate a relatively low prior
probability (low for me, high for most skeptics) of the truth of the resurrection
as a hypothesis, as well as .25 for the probability of our having the particular
sort of evidence for it that we do have (high for me, low for most skeptics), and
.4 to mean somewhat modest "predictive power" of the hypothesis, we
have:

.4 x .25

P(H│E & K) = ----------- = .4

.25

So the resurrection in this scenario would be slightly improbable.
Does this mean that we should be committed to believe with precisely 40% confidence
that the resurrection actually occurred, or to withhold belief until
probability exceeds .5? Not necessarily.

A
rational approach would surely consider not only the probability of hypotheses being true or false, but the consequences that would follow from those
hypotheses being true or false. There would seem to be nothing especially rational
about my taking a shortcut to work by crossing a bridge high over an icy lake
where the probability of its collapsing sometime within the next year is "only"
.3, for example; or moving into a neighborhood where a full 60 percent of the
residents have never been physically assaulted because it's closer to my favorite
shopping center. A good and prudent soldier does not lay down his arms whenever
the probability of his survival dips below .5 in the heat of battle (indeed, he
is far too busy to bother with such calculations, knowing that he maximizes his
probability of survival by continuing to fight). A truly rational outlook,
then, knows the difference not merely between true and false, or between
probable and improbable, but between wisdom and folly.

What this means it that a rational approach should take into
account the expected value of a given
decision, in addition to the probability of its success. Whereas expected value
is a sophisticated statistical concept, it can be defined informally for non-mathematical
sorts like me as
"the weighted average of the values that X can take on, where each possible
value is weighted by its respective probability"[1]– and
where X is a random variable, meaning a variable with different probabilities
corresponding to different possible outcomes. For example, a business owner
might use an expected value approach to determine
the more promising of two locations for a new plant, the one with the highest
expected payoff given its probability of success and its profitability if
successful.

Now let us imagine an admittedly arbitrary but finite
"payoff" scale of utils[2]to represent the level of happiness or satisfaction
that believers and skeptics should expect to receive given the truth value
of the resurrection – and given the traditional theological position that the
resurrection of Jesus ensures eternal reward for believers, eternal judgment
for unbelievers. Let's suppose that 1,000,000,000 here represents the ultimate prize
of enjoying eternal life in the kingdom of heaven, in fellowship with Jesus
Christ, along with a host of angels and redeemed believers. This signifies the reward
of believers given that the resurrection hypothesis is true. 1, on the other
hand, is the “loneliest number,” here meaning the despair of eternal judgment upon
sin in the kingdom of darkness, in subjection to Satan, along with a host of
demons and desolate unbelievers. In other words this is the complete, or virtually complete, absence of
hope, joy or satisfaction – the reward of nonbelievers given that the
resurrection hypothesis is true. In between are values representing various
less extreme levels of expected satisfaction,
depending on possible outcomes and their probabilities. In this way each decision
outcome can be assigned an expected level of reward on the scale.

For the somewhat conservative (at least for a believer like me) probability
estimate for the resurrection mentioned above, this leaves the following:

For believers, expected value of R = 1,000,000,000 x .4 = 400,000,000.Expected value of ~R for believers = +/-
500,000 x .6 = 300,000. As mentioned above, the 1,000,000,000 represents the
maximum level of satisfaction a soul can enjoy in principle. The admittedly
arbitrary number of 500,000 for the believer means here something like: "life
on earth is not too terrible, and still has its rewards – but it's not anything
like what eternal life will be." (The important thing to keep in mind here
is that this latter number is roughly equal for both believers and unbelievers.)
So on our subjective-arbitrary scale the total
expected value for believers = 400,000,000 + 300,000 = 400,300,000.

For nonbelievers, expected value for R = 1 x .6 = .6. (This is a
level of satisfaction slightly lower than the lowest on our scale, only because
the lowest whole number on the scale is multiplied by a probability of less
than one, so we can round this up to one). Expected value of ~R for unbelievers
= +/- 500,000 x .6 = 300,000 (the same as believers). Again, 1 represents the
lowest possible level of happiness a soul can experience in principle. And
for nonbelievers, the 500,000 means something like: "life on earth is not
too terrible, and still has its rewards – it may not be paradise, but it could
always be worse." So for the same arbitrary scale the total expected
value for nonbelievers = 1 + 300,000 = 300,001.

The basic idea here is that in terms of the extreme consequences
at stake, it would be more rational to accept even a slightly improbable
position of faith in Christ because of the substantially higher potential reward.
It's true that faith in Christ is costly, in that Jesus calls us to take up a
cross, deny self and follow him (Luke 9:23). We are to invest our lives in
heaven, not on earth (Matt. 6:19-21). But faith also pays compensating dividends
in the way of fruit of the Spirit and a deep sense of purpose and calling.
Circumstantially, there is no appreciable difference between the life
experiences of believers and nonbelievers. Rain falls on the just (or
justified) and the unjust (Matt. 5:45). Since life on earth has its ups and
downs, pains and pleasures, joys and heartaches, etc., for everyone, the
venture of faith, while costly, is relatively low-risk.

Of course all this is essentially a restatement of Pascal’s Wager.
Pascal developed his “wager” as something akin to the risk-reward principle
that operates in all of life, from business and finance to romance. Given the
relative scarcity of happiness on earth, the abundance of joy in the eternal kingdom
of heaven, and the considerable prospect that the Christian gospel is true, it
seems to me, as it did to Pascal, that to repent and believe in Christ would be
the wisest investment one could ever make. I would say more about Pascal and
his development of the wager, but unfortunately I am, like the rest of us,
running out of time.

[2] “In
microeconomics, happiness is measured by a concept called utility. The standard
unit of measurement that microeconomics uses to measure utility is called the util…. The util has no
concrete numerical value like an inch
or a centimeter.
It is merely an arbitrary, subjective and convenient way to assign value to
consumer choices and to measure the consumer utility or utils of one choice
against another choice.” – Marc Davis, “Microeconomics: Assumptions and Utility,”
Investopedia, www.investopedia.com/university/microeconomics/microeconomics2.asp.

Comments

DM: Does this mean that we should be committed to believe with precisely 40% confidence that the resurrection actually occurred, or to withhold belief until probability exceeds .5? Not necessarily.

If the probability is 0.4, then we can conclude the probability is 0.4. Sure, once it is higher than 0.5 we can label it as "probable", but even then if the probability is 0.6 then that meansd the probability is 0.6. It is still not rational to have a conviction that it actually happened.

Your bigger problem is that all your figures are essentially made up. In my view, this sort of approach is meaningless when it comes to miracles or any sort of one off event, as we have no way of estimating probabilities.

the problem of numbers being made up is true of any use of Bayes. I am not commenced Bayes can be used for things like the resurrection or for any historical events. I am totally against applying it to God, it violates the principle of mystical theology.

it works well for things like submarine detection and finding missing children. It doesn't work well for things that are beyond our understanding.

Pix, I only argue about probability because so many atheists assert that the probability of theism or miracles is "vanishingly low," and they use devices like Bayes' theorem in an attempt to prove the point. I don't think probability calculus is the best way to answer metaphysical and historical questions. I'm only trying to speak the language certain atheist friends seem to prefer.

In other words your point about assigning specific probabilities to singular events like miracles is well taken.

You seem to have missed my own point, however. Despite the difficulty of precisely calculating "the" probability of the resurrection, whatever it may be, if there is considerable evidence to support the resurrection (and I believe there is), the final probability of the resurrection is above zero in some proportion. In that case anyone examining the issue is faced with the practical questions of risk I was addressing. (Of course, you can always take the "Skeptical" route and simply assert that the *prior* probability is zero, in which case the posterior probability must be zero regardless of the evidence.)

DM: Pix, I only argue about probability because so many atheists assert that the probability of theism or miracles is "vanishingly low," and they use devices like Bayes' theorem in an attempt to prove the point. I don't think probability calculus is the best way to answer metaphysical and historical questions. I'm only trying to speak the language certain atheist friends seem to prefer.

I am with you and Joe here; I think those arguments are nonsense.

DM: You seem to have missed my own point, however. Despite the difficulty of precisely calculating "the" probability of the resurrection, whatever it may be, if there is considerable evidence to support the resurrection (and I believe there is), the final probability of the resurrection is above zero in some proportion. In that case anyone examining the issue is faced with the practical questions of risk I was addressing. (Of course, you can always take the "Skeptical" route and simply assert that the *prior* probability is zero, in which case the posterior probability must be zero regardless of the evidence.)

That is basically Pascal's wager.

What are the chances that I will be in a fatal car crash if I drive today? Very low. However, if we factor in the risk, suddenly the risk of going for a drive far out weighs the risk of staying at home. Conclusion? Clearly I will have a fatal car crash if I drive today, right?

Pascal's wager really has nothing to do with the resurrection. It is a question of risk vs. reward. From Wikipedia:

Pascal argues that a rational person should live as though God exists and seek to believe in God. If God does actually exist, such a person will have only a finite loss (some pleasures, luxury, etc.), whereas they stand to receive infinite gains (as represented by eternity in Heaven) and avoid infinite losses (eternity in Hell)

OK. Let's think about this. So if you want to include the resurrection in your calculation, it's not really relevant. If there is any finite probability of your belief being true (no matter how small), the risk vs, reward equation works out in favor of belief. Probability says that the poker player should take this bet.

But there's another way of approaching this calculus. Assuming that the probability of the belief being true is low, it is still true that by making this bet, you are likely to lose. Does it make sense to bet everything you have for the very slim possibility that you will get that infinite payoff?

Let's compare this to a super-lottery, where only one person on the planet will win. The payoff is having everything you ever could want. The cost of the ticket is throwing away everything you already have in your one and only life. (And I state it this way because the pursuit of a false goal would be tantamount to wasting the opportunity you have.) The poker player's calculus says go ahead and make the bet, but that is only rational in a context where the you bet doesn't cost you everything. In other words, it makes sense when you can afford to lose, but not when you can't afford to lose. If this is the one and only life you have, you really can't afford to waste it.

My favorite professor,m they guy who got me to be a historian was a math major at Brown when he got saved and went into church history. He thinks Bayes is stupid and he doesn't like because in his day(he retired years ago) mathematicians didn't like it. He always told me historical probability is never quantified.

He thinks Bayes is stupid and he doesn't like because in his day(he retired years ago) mathematicians didn't like it.

There's nothing stupid or mathematically incorrect about Bayes' Theorem. What's stupid is the way people apply it. For example, you can't state the prior probability of a specific (or one-of-a-kind) event. Statistical probability applies to a genre - something for which there is statistical data from which a probability for events of that kind can be calculated. People who try to use Bayes on things like the creation of the universe, or the existence of God are usually talking out of their ass.

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